#7168532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,37)(27,38,41,33)(28,35,29,44)(30,36,32,47)(34,46,51,43)(39,49,45,52)(40,54,42,50)(48,55,53,56);(1,2,4,5)(3,7,8,6)(11,17)(12,20)(14,21)(15,19)(25,27)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,48)(35,52)(36,50)(37,38)(43,55)(44,49)(46,56)(47,54)(51,53);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,29)(26,34,37,51)(27,39,41,45)(30,43,32,46)(33,48,38,53)(35,36,44,47)(40,55,42,56)(49,54,52,50);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,29,31,28)(26,35,37,44)(27,39,41,45)(30,43,32,46)(33,49,38,52)(34,36,51,47)(40,56,42,55)(48,54,53,50);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,43,29,46)(33,50,38,54)(34,44,51,35)(39,55,45,56)(48,49,53,52)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7075728);